Bounds for Smooth Theta Sums with Rational Parameters

Abstract

We provide an explicit family of pairs $(\alpha ,\beta )\in {\mathbb{R}}^k\times {\mathbb{R}}^k$ such that for sufficiently regular $f$, there is a constant $C>0$ for which the theta sum bound $$\left|\sum _{n\in {\mathbb{Z}}^k}f\left(\frac{1}{N}n\right)\exp \left\{2\pi i\left(\left(\frac{1}{2}\Vert n{\Vert }^2+\beta \cdot n\right)x+\alpha \cdot n\right)\right\}\right|\le C{N}^{k/2}$$ holds for every $x\in \mathbb{R}$ and every $N\in \mathbb{N}$. Central to the proof is realising that, for fixed $N$, the theta sum normalised by $N^{k/2}$ agrees with an automorphic function $|{\Theta }_f|$ evaluated along a special curve known as a horocycle lift. The lift depends on the pair $(\alpha ,\beta )$, and so the bound follows from showing that there are pairs such that $|{\Theta }_f|$ remains bounded along the entire horocycle lift.

23 pages, 3 figures